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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 | import Mathlib
import OAI.Analysis.HardSphere.Conditioning.PhysicalFullReference
namespace OAI
section
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal BigOperators
namespace HardSphere.Groups
open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital
attribute [local instance] Classical.propDecidable
theorem physicalBranch_primary_midpoint {N K M G : ℕ} {L : Length}
(hL : L.val=(K:ℝ)) (hK : 1≤K) (h50 : 50<L.val)
(μ : Measure (BridgeEndpoints N L)) [SFinite μ]
(g : Fin G) (b : Fin N→Bool) (a : GridAllocation N K (G*M))
(m : ℕ) (r R C S s₀ t : ℝ) (u : Time) (v : Fin 3→Fin K) :
((physicalBranchLaw hL μ g b a m r R C S s₀ (2*t) u).compProd
(physicalPrimaryReference hL g m t r R C S u
(fun i : {i // i∈groupUnion g a} ↦ i.val) v)).map
(physicalFullMidpoint (Function.Embedding.subtype (fun i ↦ i∈groupUnion g a))) =
(((rawPhysicalBranch hL μ g b a m r R C S s₀ (2*t) u).map
(fun z ↦ labelBridgePath N L (2*t) z midpointTime)).prod
((cellPathReference v).map (fun e ↦ projectPoint L (e midpointTime)))).map
(fun z i ↦ i.elim z.2 z.1) := by
have H := congrArg (fun ν ↦ ν.map Prod.snd)
(physicalBranch_full_midpoint_raw hL hK h50 μ g b a m r R C S s₀ t u v)
rw [Measure.map_map measurable_snd
(measurable_fst.prodMk (physicalFullMidpoint_measurable _)),
Measure.map_map measurable_snd (by
apply Measurable.prodMk
· exact (physicalDatum_measurable g m r R C S s₀ u).comp
(((labelBridgePath_measurable N L (2*t)).comp measurable_fst).prodMk measurable_const)
· apply Measurable.of_eval
intro i
cases i with
| none =>
exact (projectPoint_continuous L).measurable.comp
((continuous_eval_const midpointTime).measurable.comp measurable_snd)
| some i =>
exact (measurable_pi_apply i).comp
((continuous_eval_const midpointTime).measurable.comp
((labelBridgePath_measurable N L (2*t)).comp measurable_fst)))] at H
have hf : Measurable (fun z ↦ labelBridgePath N L (2*t) z midpointTime) :=
(continuous_eval_const midpointTime).measurable.comp (labelBridgePath_measurable N L (2*t))
have he : Measurable (fun e : C(Time,Point) ↦ projectPoint L (e midpointTime)) :=
(projectPoint_continuous L).measurable.comp (continuous_eval_const midpointTime).measurable
rw [Measure.map_prod_map _ _ hf he,Measure.map_map (by
apply Measurable.of_eval
intro i
cases i with
| none => exact measurable_snd
| some i => exact (measurable_pi_apply i).comp measurable_fst) (hf.prodMap he)]
exact H
end HardSphere.Groups
end
end
end OAI
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